How to make my prop search show the objects in my scene?
When using property search, you need to specify the collection of objects you want to search through in the 3rd and 4th parameters (in our case the list of objects in the active scene):

layout.prop_search(
    scene.ignit_panel,  # Object with your custom property
    "screen_selector",  # Your custom property
    context.scene,      # Object that contains the collection you want to search through 
    "objects"           # The collection property
)